where \(m=-u_{xx}\) , \(\sigma =\pm 1\) ,
which was recently introduced by Constantin and Ivanov in [1]}.
The variable \(u(x, t)\)  can be interpreted as
the horizontal fluid velocity and the variable \(\rho (x, t)\)  describes
the horizontal deviation of the surface from equilibrium,
all measured in dimensionless units [1]}.
